can you get pregnant about a week after period ends when having no luck before ?

Of course its possible, woman's cycle is an estimated guess for "average woman" also you may ovulate sooner than expected. 

Possible? Sure, if you actually ovulated when the sperm was still viable.  It depends on how long and regular your cycles are, and if you had sex during your ovulation window.  Most women do not ovulate that soon after their period, but who knows about you.

Also, it generally takes longer than 4-5 days from sex to have conception and then implantation. Light bleeding is not a good sign. Actually seeing implantation bleeding is rare.  It is just a couple of drops and is usually reabsorbed by the woman's body.  

The only way for you to know is to wait another couple of weeks and start testing or get a blood test from your ob/gyn. 

What you need to do is get some ovulation tests, you can buy very cheap ones from online in packs of 50 or 100, it looks like a pregnancy test and you pee on it just the same except starting a few days after your period finished once every morning, when your approaching ovulation your line will get darker ( the line will always be there but faint) once it gets darker start testing 2-3 times a day until the line is just as dark or darker than the control line and then you must start having sex. You could be ovulating at a different time than the usual 14ish day. But even if you are it’s not always easy to get pregnant and can take a full year or more to get pregnant even for healthy fertile couples. 

And for your question brown bleeding in the middle of your cycle can be normal after ovulation and not be a sign of a pregnancy, or it may be a sign of pregnancy, if it was it would be on day 6-8 after conception which is when the egg implants. I had it once and i got a tiny stabbing pain in my very lower belly which lasted a few minuets then that night I went to the toilet and when I wiped it was pink, this was day 6, it didn’t happen again on any toilet trips and my period did not arrive that month but the cramps did it felt like I could start my period any second for the whole first trimester but I never did, I had a baby girl. 

Ovulation usually happens ABOUT 14 days BEFORE the next period is due.

It is not about how long after the last period - it is more about how long until the next period.

Conception DOES NOT always happen on the day you have sex.  Sperm can live inside you for up to five days and wait for an egg to arrive.  Conception can happen within 30 minutes after sex OR as long as five days after sex happened.

Bleeding between periods can be just about anything.  Breakthrough bleeding can happen at any time for any number of reasons.  (breakthrough bleeding would be any bleeding when a period is not expected.)  At four to five days after sex, the bleeding is not implantation bleeding.

If you want to get pregnant - have MORE SEX.   Sperm can live for up to five days, but the egg only lives 12 to 24 hours.  If you want to get pregnant without trying to track for ovulation - have sex at least once every three days anytime you are not bleeding from a period.  If you don't have fertility issues - that will usually result in pregnancy within just a few months of trying.

If you aren't pregnant within one year of actively trying to become pregnant - then consult your doctor and you and your partner would be tested for fertility issues.  

Sure you can.